Theorem dfrnf 4927

Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
dfrnf.1	⊢ Ⅎ𝑥𝐴
dfrnf.2	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
dfrnf	⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem dfrnf

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrn2 4873	. 2 ⊢ ran 𝐴 = {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤}
2		nfcv 2349	. . . . 5 ⊢ Ⅎ𝑥𝑣
3		dfrnf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2349	. . . . 5 ⊢ Ⅎ𝑥𝑤
5	2, 3, 4	nfbr 4097	. . . 4 ⊢ Ⅎ𝑥 𝑣𝐴𝑤
6		nfv 1552	. . . 4 ⊢ Ⅎ𝑣 𝑥𝐴𝑤
7		breq1 4053	. . . 4 ⊢ (𝑣 = 𝑥 → (𝑣𝐴𝑤 ↔ 𝑥𝐴𝑤))
8	5, 6, 7	cbvex 1780	. . 3 ⊢ (∃𝑣 𝑣𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑤)
9	8	abbii 2322	. 2 ⊢ {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} = {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤}
10		nfcv 2349	. . . . 5 ⊢ Ⅎ𝑦𝑥
11		dfrnf.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
12		nfcv 2349	. . . . 5 ⊢ Ⅎ𝑦𝑤
13	10, 11, 12	nfbr 4097	. . . 4 ⊢ Ⅎ𝑦 𝑥𝐴𝑤
14	13	nfex 1661	. . 3 ⊢ Ⅎ𝑦∃𝑥 𝑥𝐴𝑤
15		nfv 1552	. . 3 ⊢ Ⅎ𝑤∃𝑥 𝑥𝐴𝑦
16		breq2 4054	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑥𝐴𝑤 ↔ 𝑥𝐴𝑦))
17	16	exbidv 1849	. . 3 ⊢ (𝑤 = 𝑦 → (∃𝑥 𝑥𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑦))
18	14, 15, 17	cbvab 2330	. 2 ⊢ {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
19	1, 9, 18	3eqtri 2231	1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Syntax hints:   = wceq 1373  ∃wex 1516  {cab 2192  Ⅎwnfc 2336   class class class wbr 4050  ran crn 4683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-pow 4225  ax-pr 4260

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-br 4051  df-opab 4113  df-cnv 4690  df-dm 4692  df-rn 4693

This theorem is referenced by:  rnopab  4933